WHITESBURG, Ky.—As Congress works to repeal President Barack Obama’s signature health law, Kentucky Republican Gov. Matt Bevin is already at work unwinding some of its provisions in his state.

Mr. Bevin has dismantled the state’s health-insurance exchange, moving patients to the federal website last year. He has proposed introducing new conditions for recipients of Medicaid, the federal-state health program for the poor, that would require patients to pay premiums of up to $15 a month and perform employment-related or community-service...